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Unformatted text preview: Principle of Minimum Total Potential Energy In a linear elastic body, the strain energy stored in the body due to deformation is 1 2 T V U dV σ ε = ∫ (1) and the loss of the potential energy of the applied loads for a conservative system is the negative of the work done by the loads as the structure is deformed. The strain energy for a three dimensional isotropic medium on the Cartesian coordinate system is ( 29 1 2 x x y y z z xy xy yz yz xz xz U dxdydz σ ε σ ε σ ε τ γ τ γ τ γ = + + + + + ∫∫∫ (2) Invoking the plane stress simplification adopted for a thin plate, results in ignoring , , yx xz γ γ and z σ . The constitutive law of a twodimensional medium is ( 29 ( 29 ( 29 2 2 1 1 2 1 x x y y y x xy xy E E E σ ε με μ σ ε με μ τ γ μ = + = + = + (3) The displacement components at any pointing the plate, , , , u v w may be represented in terms of the corresponding middleplane displacement components, , , , u v w by , , x y u u zw v v zw w w = = = (4) The forcedeformation relations (or kinematic relations) at any point in the plate are ( 29 ( 29 2 , , 2 , , , , , , / 2 / 2 x x x y y y xy y x x y u w u w u v w w ε ε γ = + = + = + + (5) 1 The forcedeformation relations (or kinematic relations) at any point on the plate middle plane are ( 29 ( 29 2 , , 2 , , , , , , / 2 / 2 x x x y y y xy y x x y u w u w u v w w ε ε γ = + = + = + + (6) The total strains represented with a bar are sum of the membrane strain and bending strain such that , , , 2 x x xx y y yy xy xy xy zw zw zw ε ε ε ε γ γ = + = + = + (7) Substituting Eqs. (3) into Eq. (2), gives ( 29 2 2 2 2 1 2 2 2 1 x y x y xy E U dxdydz μ ε ε με ε γ μ = + + +  ∫∫ ∫ (8) Introducing Eqs. (7) and integrating with respect to z leads to the relations m b U U U = + (9) where 2 2 2 1 2 2 2 m x y x y xy C U dxdy μ ε ε με ε γ = + + + ∫∫ (10) and ( 29 ( 29 2 2 2 , , , , , 2 2 1 2 b xx yy xx yy xy D U w w w w w dxdy μ μ = + + + ∫∫ (11) where ( 29 3 2 2 and...
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 Summer '10
 Staff
 Equations, Potential Energy, xy, Eqs.

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